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How to cite this paper: Young, F.J. (2014) Capacitance of a Three-Dimensional Interdigitated (MIM) Capacitor. Open Access
Library Journal, 1: e1167. http://dx.doi.org/10.4236/oalib.1101167
Capacitance of a Three-Dimensional
Interdigitated (MIM) Capacitor
Frederick J. Young
School of Communications and Arts, University of Pittsburgh at Bradford, Bradford, PA, USA
Received 19 October 2014; revised 24 November 2014; accepted 12 December 2014
Copyright © 2014 by author and OALib.
This work is licensed under the Creative Commons Attribution International License (CC BY).
http://creativecommons.org/licenses/by/4.0/
Abstract
The capacitance of a Metal-Ins ulator-Metal (MIM) capacitor first described in United States Letters
Patent 20070278551 (2007) is analysed in three dimensions. The finite element computer pro-
gram is given so that others may make similar calculations with their own dimensions and dielec-
tric materials. Plots of potentials and fields are presented.
Keywords
MIMS Capacitor, 3D Finite Element Solution, Voltage Rating, BaTil03 Dielectric
Subject Areas: Applied Physics, Electric Engineering
1. Capacitance of a Three-Dimensional Interdigitated (MIM) Capacitor
The geometry considered here was taken from a US Letters Patent [1]. The abstract of the invention given in the
patent follows:
“Abstract: An interdigitated Metal-Insulator-Metal (MIM) capacitor provides self-shielding and accurate
capacitance ratios with small capacitance values. The MIM capacitor includes two terminals that extend to a
plurality of interdigitated fingers separated by an insulator. Metal plates occupy layers above and below the
fingers and connect to fingers of one terminal. As a result, the MIM capacitor provides self-shielding to one
terminal. Additional shield ing may be employed by a series of additional shielding layers that are isolated f rom
the capacitor. The self shielding and additional shielding may also be implemented as an array of MIM
capacitors.”
The geometry of the capacitor considered is shown in the
0Z=
plane in Figure 1. Regions 21 and 22 are
the capacitor plates with region 21 shielding the top by being held at ground potential. Region 22 is the hot side
of the capacitor and regions 24 and 25 are part of the shielding ground. The space between regions 21 and 22 is
the dielectric. Figure 2 shows a simplified view of the capacitor to aid in setting up the geometry for the script.
Here the blue lines are the remains of region 1 being an 8 by 34 mil rectangle. Region 2 comprises barium

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Figure 1. The g eo metry given in Uni ted States Let ter s Patent 20070278551.
Figure 2. Right half of MIM with boundary conditions.
titanate (BaTi1O3) that covers the path leading from (0.3) to (1.3) to (1.29) to (5.29) to (5.3) to (7.3) to (7.31) to
(1.31) to the upper blue line at (1.34). For a distance of 1 mil to the right of this path, it is the layer of barium

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titanate. The upper compartment is pa rt of the first region of air dielectr ic. The two other compartments must b e
defined as voids so that they will not enter into the calculations. These are given by regions 3 and 4 in the script.
To find the capacitance a field calculation is necessary. The potential in the capacitor can be calculated by
solving Laplace’s equation [1] in the interior of the capacitor [2 ]. It is given by
2
0 ϕ ∇=
(1 )
subject to boundary conditions (1)
0
V ϕ =
on region 22 and (2)
0 ϕ =
on region 21. Symmetry
0n ϕ ∂ ∂=
has been invoked by default on the edges of region 1. This causes the solution to th e right-half of the XY plane
to be a mirror image of the left-half plane. The second condition is necessary for the outer boundary and is of
little importance when
2Y= −
. The Dirichlet boundar y conditions are applied in region 2 to the barium titanate
dielectric as depicted in Figure 2. Other barium titanate boundaries are held at the potentials given also in said
figure. This problem is most easily solved by extruding the geometry of Figure 2. In three dimensional
problems it is always best to inspect the various means of extrusion in order to find the simplest. Figure 3 shows
the outlines of the extrusion used here. The same condition is used on the
XY
plane at two mills above the
capacitor plates. The boundary condition on the
XY
plane at
0Z=
is
0Z ϕ ∂ ∂=
to yield symmetry about
that plane. The capacitor half-height in the Z-direction is 24 mils. Thus by symmetry one quarter of the problem
is worked. The conducting plates of the capacitor are modeled as voids. The voltage boundary conditions are
applied to region 2 which is the high dielectric filling of the capacitor. The scrip t for the so lution of this pro blem
is given below. Lines 2 and 3 set up three dimension coordinates X, Y, Z Lines 4 and 5 set the number of
contours on the graphs to 4, and cause the coordinate labels to be in mil (thousands of an inch) and the error
limit to be
0.001≦
. Lines 6 and 7 set the main variable to be V, the electrostatic potential. Lines 8 - 13 give
definitions and relate various quantities to each other. Line 9 sets the relative primitivities kdie = 8000, kair = 1,
k = kair, the ground potential
0
0V=
, the high po tential
11V=
and the permittivity of air Eps0 = 2.2489e−16
farads/mil. Line 10 establishes energy relationships. Here the stored energy density is given by
( )
2
0p
W κϕ = ∇
(2)
and the total stored energy is given by
Figure 3. The potential on the Z = (Z0 + Zth) plane in the capacitor.

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vol
ddd
p
WW XYZ=∫∫∫
(3)
Line 11 defines the capacitance in microfarads as
( )
2
10
2.0 6CeWV V=⋅−
(4)
and the electric field as
ϕ = −∇ E
(5)
and the electrostatic indu c tion as
0
κ =DE
( 6)
Line 12 set s geomet rical variables and establishes the magnitude of the electric field in all space as
mag
E=E
( 7)
In line 13 are the instructions for finding the largest magnitude of the electric field and its coordinates. Lines
14 and 15 determine the equation to be solved by finite elements. In line 15
∇⋅ =D0
( 8)
is to be executed subject to Equations (5) and (6). Extrusion of the geometry in the Z-direction begins in line 15.
Lines 16 - 24 define the surfaces on various XY planes and set their boundary conditions. In lines 25 - 39 the XY
geometry to be extruded is defined and boundary conditions are established. There the air or dielectric in each
region is specified. Line 40 allows the solution to be seen at various stages before the error is sufficiently
reduced. In line 41 various plots and reports are made and saved when the solution is finished.
2. Computer Program
DigiCap3D.pde
1. TITLE 'Three Dimension Interdigitated Capac itor Analysis'
{ The geometry for this example was taken from Metal-insulator-metal capacitors:United States Patent
20070278551.
Inventor: Anthony, Michael P. (Andover, MA, US) Link to this
page:http://www.freepatentsonline.com/20070278551.html }
2. COORDINATES
3. CARTESIAN3
4. SELECT
5. contours = 4 alias(x) = “X(mil)” alias(y) = “Y(mil)” alias(z) = “Z(mil )” errlim = 0.001
6. VARIABLES
7. V
8. DEFINITIO NS
9. kdie = 8000 kair =1 k =kair V0 = 0 V1 = 1 Eps0 = 2.2489e-16 { Farads/mil }
10. Wp = 0.5*K*eps0*grad(V)942 { Stored Energy Density } W = vol_integral(Wp) { Total Stored
Energy }
11. C = 1.0e6*2*W/(V1-V0)942 { Capacitance in microFarads } E = -grad(V) D = K*Eps0*E
12. Cth = 2 Dth = 1 Xmax = 2.5*Cth+3*Dth Ymax = 34 Z0 = 2 Zth = 24 Emag = magnitude(E)
13. Emax = globalmax(Emag) XX = globalmax_x(Emag) YY = globalmax_y(Emag) ZZ =
globalmax_z(Emag)
14. EQUATIONS
15. DIV(D) = 0
16. EXTRUSION
17. SURFACE “Bottom” Z = 0
18. LAYER “Dielectric”
19. SURFACE “Top Metal - Dielectric” Z = Zth
20. LAYER “Air”
21. SURFACE “Top” Z= Z0 + Zth

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22. BOUNDARIES
23. SURFACE “Bottom” natural(V)=0
24. SURFACE “Top” natural(V)=0
25. BOUNDARIES
26. REGION 1 'Universe'
27. surface “Bottom”
28. layer 'Dielectric' k = kair
29. start(0,Cth) line to (Xmax,Cth) to (Xmax,Ymax) to (0,Ymax) to close
30. REGION 2 'Dielectric'
31. SURFACE 'Top Metal - Dielectric'
32. LAYER “Dielectric” k = kdie
33. start(0,Cth+Dth) valu e(V) = V1 line to (Cth/2,Cth+Dth) to (Cth/2,Ymax-2*Cth-Dth) to
(2*Dth+1.5*Cth,Ymax-2*Cth-Dth) to (2*Dth+ 1.5*Cth,Cth+Dth) to (2*Dth+2.5*Cth,Cth+ Dth) to
(2*Dth+2.5*Cth,Ymax-Cth-Dth) to (Cth/2,Ymax-Cth-Dth) to (Cth/2,Ymax) natural(V)=0 line to
(Dth+Cth/2,Ymax) value (V)=V0 lin e to (Dth+Cth/2,Ymax-Cth) to (3*Dth+2.5*Cth,Ymax-Cth) to
(3*Dth+2.5*Cth,Cth) to (Dth+1.5*Cth,Cth) to (Dth+1.5*Cth,Ymax-2*Cth-2*Dth) to
(Dth+Cth/2,Ymax-2*Cth-2*Dth) to (Dth +Cth/2,Cth) to (0,Cth) natural(V) = 0 line to close
34. REGION 3
35. SURFACE 'Top Metal - Dielectric'
36. layer “Dielectric” vo id
37. start(0,Cth+Dth) line to (Cth/2,Cth+Dth) to (Cth/2,Ymax-2*Cth-Dth) to
(2*Dth+1.5*Cth,Ymax-2*Cth-Dth) to (2*Dth+ 1.5*Cth,Cth+Dth) to (2*Dth+2.5*Cth,Cth+ Dth) to
(2*Dth+2.5*Cth,Ymax-Cth-Dth) to (Cth/2,Ymax-Cth-Dth) to (Cth/2,Ymax) to (0,Ymax) to close
38. start (Dth+Cth/2,Cth) line to (Dth+Cth/2,Ymax-2*Cth-2*Dth) (Dth+1 .5 *C t h,Ymax-2*Cth-2*Dth) to
(Dth+1.5*Cth,Cth) to clo s e
39. start(Xmax,Cth) lin e to (Xmax,Ymax) to (Dth+Cth/2,Ymax) to (Dth+Cth/2,Ymax-Cth) to
(3*Dth+2.5*Cth,Ymax-Cth) to close
40. MONITO R S
contour(V) on Z = (Z0 + Zth)/2 contour(V) on X = 0 vector(E) on Z = (Z0 + Zth)/2 report C
41. PLOTS
grid (x,y,z) grid(x,y) on Z = Z0+Zth grid(y,z) on X = 3 contour(V) on Z = (Z0 + Zth)/2 report(C)
report(Emax)
vector(E) on Y = 3 as 'Electric intensity' v ec tor (E) on Y = Ymax/2 as 'Electric intensity'
vector(E) on X = 2*(Dth+Cth) zoom(0 ,17,10,10) as 'Electric intensity'
vector(E) on Z = Z0 zoom(Dth+2.5*Cth,Ymax-Cth-2*Dth,2*Dth,2*Dth) as 'Electric intensity'
vector(E) on Z = (Z 0 + Z t h) /2 zoom( Dt h+2.5*C t h,Ymax-Cth-2*Dth,2*Dth,2*Dth) as 'Electric intensity'
vector(E) on Z = Zth/2 zoom(Dth+2.5*Cth,Ymax-Cth-2*Dth,2*Dth,2*Dth) as 'Electric in te nsity'
vector(E) on Z =13.8851 zoom(XX-0.05 ,YY-0.05,0.1,0.1) report(Emax) surface(E mag) on Z = 13.8851
vector(E) on z = (Z0 + Zth)/2 zoom(0,26,8,8) as 'Electric intensity' vector(E) on z = (Z0 + Zth)/2
zoom(0,1,8,8) as 'Electric intensity'
report(C) as 'Total capacitance in ufarads'
42. SUMMARY
Report(C) as 'Total Capacitance in ufarads'
REPORT(C/Zth) as “Capacitance(u F/mil in z dimension)”
REPORT(W) as “Stored Energy”
REPORT(Emax) as 'The maximum electric intensity'
REPORT(XX) as 'The X-coordinate of the maximum electric inten s ity'
REPORT(YY) as 'The Y-coordinate of the maximum electric intensity'
REPORT(ZZ) as 'The Z-coordinate of the maximum electric inten s ity'
END
3. Results
Note the aforementioned symmetry conditions under boundaries. The total field energy is calculated by

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integrating over the entire volu me. From that the cap acitance is calculated for this quarter of the whole capacitor.
The dimensions of various parts of the device are in literal form so that different designs may be done easily.
The global max function is applied to the magnitude of the electric intensity vector to determine where the
largest value occurs. Figure 3 shows the potential distribution in the dielectric of the capacitor on an internal Z
plane. It is important to check this figure to make sure that the correct boundary conditions on potential have been
used. The Dirichlet conditions have been met and the symmetry condition required at
0X=
has been met for
23Y≤≤
or in more general terms
thth th
CYC D≤≤ +
. The equipotentials are crowded near some of the
corners, indicating larger electric intensities there than elsewhere. In Figure 4 and Figure 5 the vector plots
of the upper and lower corners in the
( )
0
2
th
ZZZ= +
plane are shown. In the region not exhibited the
electric intensity is almost co ns tant as it would be in a simple one dimensional parallel plate cap a citor.
The electric intensity is important because it must not exceed the breakdown electric intensity of the dielectric
material. Previous calculations would indicate that the electric intensity may be greatest near sharp edges or
corners. Figure 6 shows the electric in tensity on the
13.8851Z=
plane where the electric intensity reaches its
maxima. It is sometimes difficult to present the E vectors on the correct plane to capture the largest electric
intensity vector in a plot.
The script for solving this problem contains many more plots of the mesh and vectors at various places. The
summary yields total capacitance of
3
4.06 10μ
−
×
farads and a maximum electric intensity of 3.97 volts/mil at
the coordinates (2.28, 13.8851). An estimate of the capacitance is obtained by expressing the plate area as
meanp th
AlZ=
(9 )
where
p
A=
the plate area,
mean
l=
the mean length of the dielectric in the
X
and
Y
directions and
th
Z=
the height of the plates. This calculation which neglects fringing estimates the total capacitance to be
3
4.19 10μ
−
×
farads. For the entire capacitor these values of capacitance must be multiplied by four because of
the symmetry used in this script. In this geometry the greatest value of a three-dimensional calculation is
prediction of where the largest value of electric intensity occurs. To actually see the largest electric intensity
vector would require the calculation of more electr ic intensity plots and is left as a task for the interested reader.
For every dielectric there exists a value of electric intens ity, called the dielectric breakdown str ength, which will
Figure 4. Electric intensity vectors on the Z = (Z0 + Zth)/2 plane .

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Figure 5. Electric intensity in the vicinity of its maxima.
Figure 6. The magnitude of the electric intensity on the Z = 13.89 plane.
result in arcing and dielectric damage. The exact value depends upon temperature, density and material com-
position. The variation of relative permittivity with material composition is given by V. A. Russel [3]. For a
dielectric containin g 65 % BaTiO3 and 35% SrTiO3 the breakdown strength of the dielectric is given in Table 1.

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Table 1. Dielectric break down voltage versus density.
Density (g/cc) Breakdown (volts/mil)
5.41 122
5.44 142
5.46 190
5.56 291
5.57 295
5.59 350
Table 1 indicates that the breakdown voltage can be as large as 350 volts/mil for the densest dielectric. In that
case the capacitor under consideration could operate with an applied voltage less than
350 3.9788.16=
volts
without being damaged. In Figure 6, the surface of the magnitude of the electric intensity on the
13.8851Z=
plane is exhibited. The sharp peaks in the magnitude of the electric intensity in corners are rather large compared
to the value of the electric intensity in th e rest o f th e dielec tric.
References
[1] Anthony, M.P. and Andover, M.A. (2007) Meta l-Insulator-Metal Capacitors. US Patent 20070278551.
http://www.freepatentsonline.com/20070278551.html
[2] Panofsky, W.K.H. and Phillips, M. (1955) Classical Electricity and Magnetism. Addison-Wesley Publishing Company,
Inc., Cambridge, Mass.
[3] Russel, V.A. (1962) Ceramic Material of High Dielectric Strength Containing Barium Titanate and Method of Manu-
facture. US Patent 3049431. http://www.freepatentsonline.com/3049431.pdf